In
2012, Gutman and Wagner proposed the concept of the matching energy of a graph
and pointed out that its chemical applications can go back to the 1970s. The
matching energy of a graph is defined as the sum of the absolute values of the
zeros of its matching polynomial. Let u and v be the non-isolated vertices of
the graphs G and H with the same order, respectively. Let wi be a non-isolated
vertex of graph Gi where i=1, 2, …, k. We use Gu(k) (respectively, Hv(k)) to denote the graph which is the coalescence of G (respectively, H) and G1, G2,…, Gk by identifying the
vertices u (respectively, v) and w1, w2,…, wk. In this paper, we first present a new technique of directly
comparing the matching energies of Gu(k) and Hv(k), which can tackle some quasi-order incomparable problems. As
the applications of the technique, then we can determine the unicyclic graphs
with perfect matchings of order 2n with the first to the ninth smallest matching energies for all n≥211.

Let G be a simple and undirected graph with n vertices and
A
(
G
)
be its adjacency matrix. Let
λ
1
,
λ
2
,
⋯
,
λ
n
be the eigenvalues of
A
(
G
)
. Then the energy of G, denoted by
E
(
G
)
, is defined as [1]

E
(
G
)
=
∑
i
=
1
n
|
λ
i
|
.

A fundamental problem encountered within the study of graph energy is the characterization of the graphs that belong to a given class of graphs having maximal or minimal energy, for example, Trees with extremal energies [2] - [15] ; Unicyclic graphs with extremal energies [16] - [21] ; Bicyclic graphs with extremal energies [22] [23] [24] [25] ; Tricyclic graphs with extremal energies [26] [27] [28] . For more details, they can be found in the recent book [29] and review [30] .

A matching in a graph G is a set of pairwise nonadjacent edges. A matching is called k-matching if its size is k. Let
m
(
G
,
k
)
be the number of k-matching of G, where
m
(
G
,
k
)
=
0
for
k
>
⌊
n
/
2
⌋
or
k
<
0
. In addition, we assume that
m
(
G
,
0
)
=
1
.

In [31] , Gutman and Wagner shown that its matching energy coincides with its energy if T is a forest. Many properties of the matching energy are analogous to those of the graph energy. However, there are some notable differences. Then they raised a question: is it true that the matching energy of a graph G coincides with its energy if and only if G is a forest? Up to now, the question is still open.

The study on extremal matching energies is very interesting. In [31] , Gutman and Wagner characterized the unicyclic graphs with the minimal and maximal matching energy. Zhu and Yang [35] determined the unicyclic graphs with the first eight minimal matching energies. In [36] , Chen and Liu characterized the bipartite unicyclic graphs with the first
⌊
(
n
−
3
)
/
4
⌋
largest matching energies. Moreover, Chen et al. [37] determined the unicyclic odd-cycle graphs with the second to the fourth maximal matching energies. For bicyclic graph, Ji et al. [38] obtained the graphs with the minimal and maximal matching energy. In [39] , Liu et al. further determined the bicyclic graphs with first five minimal matching energies and the second maximal matching energies, respectively. Chen and Shi [40] characterized tricyclic graph with maximal matching energy, for more results about extremal matching energies, see [41] - [47] .

A fundamental problem encountered within the study of the matching energy is the characterization of the graphs that belong to a given class of graphs having maximal or minimal matching energy. One of the graph classes that are quite interestingly studied is the class of all unicyclic graphs with perfect matchings. As far as we are concerned, no results are on this topic. In this paper, we first present a new technique of directly comparing the matching energies of
G
u
(
k
)
and
H
v
(
k
)
in Section 2 (see Figure 2). As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all
n
≥
211
in Section 3.

For simplicity, if
G
1
is isomorphic to
G
2
, then we write
G
1
=
G
2
. If
G
1
is not isomorphic to
G
2
, then we write
G
1
≠
G
2
. Let
A
(
2
n
)
be the set of the unicyclic graphs with perfect matchings of order 2n. Let the unicyclic graphs
A
1
,
A
2
,
A
3
,
A
4
,
A
4
*
,
A
5
,
A
6
,
A
7
,
A
8
,
A
9
be shown in Figure 1. The following theorem is the main result of this paper.

2. A New Technique of Directly Comparing the Matching Energies of
G
u
(
k
)
and
H
v
(k)

By Definition 1.2, we can see that the quasi-order method can be used to compare the matching energies of two graphs. However, if the quantities
m
(
G
,
k
)
cannot be compared uniformly, then the common comparing method is invalid, and this happens quite often. Recently much effort has been made to tackle these quasi-order incomparable problems [35] [39] [40] .

Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let
w
i
be a non-isolated vertex of graph
G
i
where
i
=
1
,
2
,
⋯
,
k
. We use
G
u
(
k
)
(respectively,
H
v
(
k
)
) to denote the graph which is the coalescence of G (repectively, H) and
G
1
,
G
2
,
⋯
,
G
k
by identifying the vertices u (respectively, v) and
w
1
,
w
2
,
⋯
,
w
k
(see Figure 2). In [14] , He et al. presented a new method of directly comparing the energies of the bipartite graphs
G
u
(
k
)
and
H
v
(
k
)
. In this section, we apply the main idea of this method to present a new technique of comparing the matching energies of the graphs
G
u
(
k
)
and
H
v
(
k
)
which can be used to tackle these quasi-order incomparable problems.

The coalescence of two graphs G and H with respect to vertex u in G and vertex v in H, denoted by
G
u
⋅
H
v
(sometimes abbreviated as
G
⋅
H
), is the graph obtained by identifying the vertices u and v. Zhu and Yang [35] shown the recurrence relation of
α
˜
(
G
⋅
H
)
in the following. For convenience of the reader, we present a full proof.

Lemma 2.3. ( [35] ) Let
G
⋅
H
be the coalescence of two graphs G and H with respect to vertex u in G and vertex v in H. Then

which implies that the result holds. We assume that the result holds for
k
−
1
in what follows. For simplicity, we write
h
k
=
∑
i
=
1
k
α
˜
(
G
i
)
α
˜
(
G
i
−
w
i
)
−
k
x
. By Lemmas 2.1 and 2.3, we can show

Next, we use the new technique to compare the matching energies of the quasi-order incomparable graphs
A
5
and
A
6
,
A
8
and
A
9
(see Figure 1), respectively. Denote by
C
k
and
P
k
the cycle of length k and the path of length
k
−
1
, respectively.

Lemma 2.6. If
n
≥
6
, then
M
E
(
A
5
)
<
M
E
(
A
6
)
.

Proof. Let G be the graph obtained by attaching a pendent edge to a vertex u of
C
5
. Let H be the graph obtained by attaching a pendent edge and a pendent path of length 2 to the vertices w and v of
C
3
, respectively. Let
G
1
=
G
2
=
⋯
=
G
n
−
3
=
P
3
and
w
i
be the pendent vertex of
G
i
. Then
G
u
(
n
−
3
)
=
A
5
and
H
v
(
n
−
3
)
=
A
6
(see Figure 1). By some calculations, we can show

Proof. Let G be the graph obtained by attaching two pendent paths of length 2 to the same vertex of
C
4
. Let H be the graph obtained by first attaching a pendent edge to each vertex of
C
3
and then attaching a pendent path of length 2 to one vertex of
C
3
. Let u be the vertex of degree 4 in G and v be the vertex of degree 3 in H, respectively. Let
G
1
=
G
2
=
⋯
=
G
n
−
4
=
P
3
and
w
i
be the pendent vertex of
G
i
. Then
G
u
(
n
−
4
)
=
A
8
and
H
v
(
n
−
4
)
=
A
9
(see Figure 1). By some calculations, we can get the followings.

3. Minimal Matching Energies of Unicyclic Graphs with Perfect Matchings of Order 2n

In this section, we will determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies (i.e., to prove Theorem 1.1).

In what follows, we denote by
M
(
G
)
a perfect matching of a graph G. Let
G
^
=
G
−
M
(
G
)
−
S
0
, where
S
0
is the set of isolated vertices in
G
−
M
(
G
)
. We call
G
^
the capped graph of G and G the original graph of
G
^
. For example, the capped graphs of
A
1
,
A
2
,
A
3
,
A
5
are shown in Figure 3.

This is the main method to compute
m
(
G
,
k
)
of a graph G in what follows.

Figure 3. The capped graphs of
A
1
,
A
2
,
A
3
and
A
5
. For each graph, the dashed lines denote the copies of
P
2
attached to the maximal degree vertex.

Let
X
n
be the star of order n. Let
Y
n
be the graph of order n obtained by attaching
n
−
3
pendent edges to a pendent vertex of
P
3
. Let
Z
n
be the graph of order n obtained from
P
4
=
v
1
v
2
v
3
v
4
by attaching
n
−
5
and one pendent edges to
v
2
and
v
3
, respectively. In [2] and [31] , the following results were shown.

Lemma 3.2. ( [31] ) Suppose that G is a connected graph and T is an induced subgraph of G such that T is a tree and T is connected to the rest of G only by a cut vertex v. If T is replaced by a star of the same order, centered at v, then the quasi-order decreases (unless T is already such a star).

Let
S
n
l
be the unicyclic graph of order n obtained by attaching
n
−
l
pendent edges to one vertex of
C
l
.

It implies that
A
1
≺
A
2
≺
A
3
≺
A
4
≺
A
5
and
A
6
≺
A
7
≺
A
8
. From Lemmas 2.6 and 2.7, the result can be easily obtained.

Proof of Theorem 1.1:

Proof. The result can follow immediately by Lemmas 3.13 and 3.14.

4. Conclusions

In this paper, we first present a new technique of directly comparing the matching energies of
G
u
(
k
)
and
H
v
(
k
)
, which can tackle some quasi-order incomparable problems. As the applications of the technique, we then determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all
n
≥
211
. Furthermore, we can consider characterizing the extremal graphs with maximal or minimal matching energy for other classes of graphs, e.g. graphs with different parameters. These are our work in the future.

The results presented in this paper are for a fixed graph. In reality, most of the graphs or networks are evolving. Some graph invariants have been studied in this setting, e.g. the Estrada index of evolving graphs [48] ; Laplacian Estrada and normalized Laplacian Estrada indices of evolving graphs [49] . Then we can consider studying the matching energy of evolving graphs in the future.

Acknowledgements

We thank the editor and the referee for their valuable comments. This work is supported by the National Natural Science Foundation of China (No. 11501356) and (No. 11426149).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.